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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1071 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 31624. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1071.7 | ⊢ 𝐷 = (ω ∖ {∅}) |
Ref | Expression |
---|---|
bnj1071 | ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1071.7 | . . 3 ⊢ 𝐷 = (ω ∖ {∅}) | |
2 | 1 | bnj923 31384 | . 2 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω) |
3 | nnord 7334 | . 2 ⊢ (𝑛 ∈ ω → Ord 𝑛) | |
4 | ordfr 5978 | . 2 ⊢ (Ord 𝑛 → E Fr 𝑛) | |
5 | 2, 3, 4 | 3syl 18 | 1 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ∖ cdif 3795 ∅c0 4144 {csn 4397 E cep 5254 Fr wfr 5298 Ord word 5962 ωcom 7326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-tr 4976 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-om 7327 |
This theorem is referenced by: bnj1030 31601 bnj1133 31603 |
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